Download Fibonacci-Tribonacci - The Fibonacci Quarterly

FIBONACCI-TRIBONACCJ
70
[ O c t 1963]
Editorial Comment
Mark Feinberg is a fourteen-year-old student in the ninth grade of the Susquehanna Township Junior High School and recently became the Pennsylvania State
Grand Champion in the Junior Academy of Science, This paper is based on his
winning project and is editorially uncut. Mark Feinberg, in this editor f s opinion, will go far in Ms chosen field of endeavor. Congratulations from the editorial staff of the Fibonacci Quarterly Journal, Mark!
34
13
21 .
13
8
,?
Golden Rectangle
SilverH Rectangle
Figure 1
13
7
4
2
?
f
Fibonacci?? Tree
Figure 3
Tribonacci n Tree
FIBONACCI-TBIBONAOCI*
MARK FEINBERG
For this Junior High School Science Fair project two variations of the
Fibonacci series were worked out.
"TRIBONACCF
Just as in the Fibonacci series where each number is the sum of the preceding twos or p
.. = p
+ p _.., the first variation is a series in which each
number is the sum of the preceding three, or q
n
the series is called Tribonaccie
n
.. = q
+
Q
ral
+Q
o ; hence
Its first few numbers are
1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504 • • •
Like the Fibonacci series, the Tribonacci series is convergent
the Fibonacci fractions p /(P
+1
and p
/p
Where
converge on .6180339-—
and
1,6180339... , the Tribonacci fraction of any number of the series divided by
the preceding one (q /q
n
convergents are termed Phi
f?
Tri-Phi
n
) approaches ,54368901* ** . While the Fibonacci
v?
(<p)5 the Tribonacci convergents might be called
(03).
Series-repeating characteristics are shown in the famed Fibonacci Golden
Rectangle.* A rectangle can be made of the Tribonacci series which also has
series-repeating characteristics but since they are less obvious this rectangle
might be called the "Silver Rectangle. M Its length (q
) and its width (q )
make it proportionately longer than the Golden Rectangle.
By removing the squares q
by q
and q __1 by q
- , two new rec-
tangles in the proportion of the original appear (shaded areas).
by q
series.
One is q _ r
o ; but the other is composed of numbers not found in the Tribonacci
This rectangle is (q , ' - q ) by (q - q n ) and is formed of numx
t>
^n+l
n' J
n TI-1
bersfrom an intermediate series obtained by subtracting each Tribonacci number from the one after it,
*See editorial remarks, page 70.
Figure 1 appears on page 70.
71 •
FIBONACCI-TRIBONACCI
72
[Oct.
By carrying the rectangle out farther new numbers found in neither the
original Tribonacci series nor the intermediate series appear. These are of a
second intermediate series and are obtained by subtracting each number of the
first intermediate series from the succeeding one. New numbers of new intermediate series also appear by further carrying out the rectangle. These other
series are formed by triangulating in the same way as the first two intermediate series. All these intermediate series are convergent upon the
n
Tri-Phi,!
values and each number in each of these series is the sum of the preceding
three.
Figure 2 shows the first two intermediate series.
1,
0,
1,
1,
1,
1,
•!,. 3, 5,
9,
17, 31 '•••.
3, 6, 11, 20, 37, 68 •••
4, 7, 13, 24, 44, 81, 149 •••.•'
2,
2,
Figure 2
The two Fibonacci convergents fit the quadratic equation x = 1 + l / x .
The Tribonacci convergent of any number in the series divided by the preceding
one (q
- / q ) fits the cubic equation y = 1 + l / y + l/y 2 . It is derived thus:
The formula giving any number in the series is
Vi = V + V i
Dividing by
+
V2 •
q^:
Vi
Vi
=
+ 1 +
Vi
V2
Vi
Let
V l 1=
q
n
V
—s- = t ,
n 1 ' V2
Vl
~
n 2
~
Then since
Vi
Vi
Vi "
\
Vi
v~ %
A_ . t
Vi
n
h-1
1963 ]
73
FIBONACCI-TRIBONACCI
Therefore
t
• t
n
- =
n-1
q
+ 1 +
q
1
TI~1
t
n
• t
1
TI-1
Substituting for the r e s t of the formula:
1
- = t , + 1 + .
n-1
n-1
t
0
n-2
Dividing& by
t
J
- :
n-1
t
All the t
n
x
= 1 + ^ +
t .,
n-1
t
.. • t 0
n-1
n-2
t e r m s converge upon one value
substituted for all t
n
terms.
(y).
Therefore
f! n
y
can b e
So
y = l
+
.
U±
y
y2
The convergent approached by any n u m b e r of the s e r i e s divided by the
succeeding one (q
/q
- ) fits the cubic equation 1/y = 1 + y + y 2 and i s d e -
r i v e d through a s i m i l a r p r o c e s s .
Charting the Fibonacci convergent .6180339- • • on p o l a r coordinate p a p e r
i s known to p r o d u c e the famed s p i r a l found all over n a t u r e .
By c h a r t i n g the
T r i b o n a c c i convergent .54368901- • • a slightly t i g h t e r s p i r a l i s produced.
It i s not known whether the T r i b o n a c c i s e r i e s has any n a t u r a l applications.
A well-known Fibonacci application i s of a hypothetical t r e e . If each limb w e r e
to s p r o u t another l i m b one y e a r and r e s t the next, the n u m b e r of l i m b s p e r y e a r
would total 1, 2, 3, 5, 8- • • in Fibonacci sequence.
However if e a c h
limb
on the t r e e w e r e to s p r o u t for two y e a r s and r e s t for a y e a r , the n u m b e r of
l i m b s p e r y e a r would total 1, 2, 4, 7, 13 • • • in T r i b o n a c c i sequence.
See
F i g u r e 3, page 70.
Could such a t r e e as that on the r i g h t be c a l l e d a
n
Tree-bonacci?T?
"TETRANACCI"
The second v a r i a t i o n of the Fibonacci sequence i s a s e r i e s in which each
n u m b e r i s the s u m of the p r e c e d i n g four n u m b e r s o r r , ^ = r + r
.,+r
0
&
n+1
n
n-1
n-2
[ O c t 1963]
FIBONACCI-TRIBONACCI
74
+r
.
n-3n
are:
Therefore this series is called "Tetranacci.
M
Its first few numbers
1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773 •• •
.
Like the Fibonacci and Tribonacci series, the Tetranacci series is convergent.
The fraction r
- /r
converges upon 1.9275619-•• and fits the
fourth power equation z = 1 + l / z + l / z 2 + l / z 3 .
The fraction r
IT
2
.. converges upon .51879006- •• which fits the equa3
tion l / z = 1 + z + z + z .
The derivation of these formulas follows the same algebraic process as
that given above and will be gladly furnished upon request.
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